Controlled experimental research and model design of double-layer high-frequency vibrating screen machine

The double-layer high-frequency vibrating screen machine is widely used in the tobacco industry, and its controllability of vibration has a great influence on the screening quality of cut tobacco. In this paper, we obtain the factors affecting the motion characteristics of the vibrating screen by theoretical derivation and experimental measurement of the motion parameters of the double-layer high-frequency vibrating screen machine, and determine the relationship between various factors and the amplitude of the vibrating screen. We also studied the properties of material motion. Through the overall research, we found that in the controllable process of realizing the high-frequency double-layer vibrating machine, we need to pay attention to the quality of the rotor of the motor, the angular velocity, and the radius of gyration, and also need to consider the number of spring piece and the rigidity of the spring piece. Also, the quality of the material will affect the movement of the screen.


Introduction
In the 16th century, the United Kingdom, Germany, the United States, and other countries began to study the screening technology [1,2], and now its vibrating screen equipment and theoretical technology are at a relatively high level. Compared with the level of foreign industrial technology, the domestic previous research on the vibrating screen is still relatively backward [3][4][5]. At present, researchers at home and abroad have adopted CAD technology, advanced manufacturing technology, optimized design methods, and other means to develop a variety of specifications, wide applicability, excellent performance of new screening equipment [6][7][8][9]. However, Guo Nianqin [10] believes that foreign vibrating screens have occupied the high-end market of vibrating screens in China, and there are few differences in the design frequency, amplitude, exciting force, and other main parameters between domestic and foreign vibrating screens. The main distinctions between the advantages and disadvantages of the vibrating screens are the rationality of the structural design of the body, the reliability of the exciter, the advanced processing technology, and the long-term effectiveness of the body material [11,12]. Vibrating screens are critical machines prone to successive failures that can result in huge economic losses [13] and must be constantly improved in order to meet the requirements of the mining industry [14]. At present, the University of Concepción, Edmundo Larenas has used two 2D three-degreesof-freedom dynamic models for a vibrating screen are tested, using linear and nonlinear approaches for angular displacement. Tests and mechanical modeling have proved that the linear dynamic model is sufficient to evaluate the motion characteristics of the double-layer vibrating screen [15], which is consistent with the method used in this article. The law of change between various parameters is not considered. Dynamic models of vibrating screens can simulate motion of the vibrating screen structure and show good agreement with experimental measurements [16] and finite element method (FEM) results [17]. The linear model considers three degrees of freedom. The excitation force is circular in the vertical direction, and the vibrating screen structure is supported by different damping springs stiffnesses in each support position. Our model calculated the dynamical response and natural frequencies, and the optimization used FEM and experimental measurements.
Because of the current high standard and diversification of tobacco classification and screening, multi-layer tobacco vibrating screen has become the mainstream of research and development [18,19]. Based on the research of the dual-mass high-frequency vibrating screen independently developed by Qinhuangdao Tobacco Machinery Factory, the factors affecting the motion characteristics of the double-layer high-frequency vibrating screen are obtained through the theoretical derivation of the motion parameters and field vibration test, and the relationship between the factors and the amplitude of the vibrating screen is determined. Currently, there is no design specification for the tobacco vibrating screen involved in this project in China [20]. The design theory can only be manufactured by imitating foreign equipment such as Germany, lack of core design theory, and parameters of the controllable data. Therefore, the products produced can only rely on experience [21], often fail to meet the desired design requirements. The controllable parameters of the design of the vibrating screen body are obtained by theoretical and experimental research on the controllable design of the two-way amplitude of the upper and lower screens and the bottom plate of the double-layer high-frequency vibrating screen machine, combining the physics conservation of momentum and the two-degree-of-freedom spring-mass system of vibration mechanics, and the field test analysis and demonstration are carried out. The purpose of our research is to systematically analyze the influence of the model parameters of each component of the double-layer vibrating screen on the vibration characteristics of the system, and realize the design and control of each vibration parameter according to different application performance requirements.    Fig. 1, two motors of the same specification do synchronous reverse motion on the lower insulator. The structure of the eccentric block inside the motor is shown in Fig. 2. To ensure the normal operation of the vibrating screen machine, the eccentricity and mass of the eccentric block in the motor are equal, respectively. The coordinate system shown in the figure is established, and the -axis is the axis of symmetry passing through the mass center of the vibrating screen. According to the force analysis, the instantaneous centrifugal inertia force generated by the rotation of the eccentric block is always in the opposite direction along the -direction, offsetting each other. In the direction of the -axis, the forces are superposed with each other, forming a single direction excitation force.

Establishment of the mechanical model
Under the motion of the eccentric block of the motor, the magnitude of the exciting force changes periodically, which realizes the reciprocating linear motion of the vibrating screen. When the vibrating screen operates at any time , the calculation formula of its exciting force value is as follows: where: -exciting force in the direction; -maximum exciting force; -the sum of eccentric Block Mass on Vibrator; -rotation radius of eccentric block; -rotation angular velocity of exciter.
The amplitude and frequency of the motor's exciting force are the vibration sources that make the double-layer or even three-layer vibrating screen produce forced vibration. According to the transmission speed and screening quality requirements of the tobacco leaf, the motor speed can be adjusted and the horizontal movement speed of the vibrating screen can be controlled. Can stimulate the vibration of the vibrating machine with a larger amplitude according to the design requirements.

Global analysis based on momentum conservation
Eq. (1) gives the calculation formula of an exciting force. When discussing momentum, we first study the impulse. The impulse is a vector, describing the accumulation effect of force on time. The expression is: In the case studied in this paper, the exciting force is a function of time and periodic variation, so in this case, the impulse should be expressed as: After calculation, it is easy to obtain that the integral shown in Eq. (3) is 0. It can be seen that the impulse of exciting force in a period is 0. The impulse is 0, which means that the momentum of the system does not change in a period, that is, momentum conservation.
The expression of momentum is: where: -total vibration mass of vibrating body and material; -moving speed of vibrating body and material. In a period, the product of and should change periodically when is unchanged, which means that the larger the total mass of the vibrating body and the material, the smaller the corresponding velocity value.

Analysis of the influence law of the rigidity and quality of the double-layer screen body
If the system is composed of particles, let the mass of the th particle , the radius , the active force acting on the particle, the restraining reaction , and the inertial force , which can be obtained by using the principle of virtual work: Then the general equation of dynamics under ideal constraint is: ∑ ( + ) ⋅ = 0 Among them, the virtual work of the active force is: -generalized hypocrisy. The virtual work of inertia force is: where: -kinetic energy of the system. By substituting the virtual work of active force and inertia force into Eq. (5), the Lagrange equation of non-conservative system can be obtained as follows: The motion of the vibrating screen can be regarded as a steady forced vibration composed of mass elastic-element-damping, in which mass refers to the weight of the vibrating screen equipment, elastic element refers to the damping spring device between the vibrating screen and the main building of the equipment, and damping includes spring damping and air damping. According to the theory of mechanical vibration, this kind of dynamic system can be regarded as forced vibration in an independent coordinate system.
The establishment of the differential equation of motion of the dual-mass high-frequency vibrating screen. In this paper, the above-mentioned dual-mass high-frequency vibrating screen is regarded as a vibration system with three degrees of freedom, and its model is established as shown in Fig. 3: where: -the mass of the eccentric block of the motor; -rotation radius of the eccentric block; -excitation frequency; -angle between motor and horizontal plane: By substituting and into Eq. (7), the differential equations of motion of the dual-mass high-frequency vibrating screen can be obtained: The specific characteristic equation can be obtained as follows: Among them, = 320000 N/m, = 1480000 N/m, = 55°, = 870 kg, = 300. According to the test results, the rigidity of the spring pieces can be obtained = 34000×38 = 1292000 N/m. By substituting the data, the stiffness matrix and mass matrix can be obtained as: Matlab is used to solve the problem, and the first to third order natural frequencies are obtained as shown in Table 1. The excitation frequency of the vibrating screen is 78.54 rad/s. From Table 1, it can be seen that the third-order natural frequency of the vibrating screen is lower than the excitation frequency, so the resonance of the high-frequency vibrating screen will not occur under the normal working frequency. However, in the process of starting and braking, the motor will pass through the resonance area of the vibrating screen, which makes the screen body shake greatly in the process of starting and braking. Therefore, we must carry out transient analysis of the screen body starting the process and study the stress changes of each part of the screen body in the starting process.

Change curve of natural frequency of vibrating screen with various physical parameters
As shown in Fig. 4, it is a regular curve of the natural frequency of each order of the high-frequency vibrating screen changing with the stiffness , that is, a regular curve of the natural frequency of each order with the number of upper-layer spring pieces. It can be seen from the figure that when increased from 1.02×e6 N/m to 2.04×e6 N/m, the number of spring pieces increased from 30 groups to 60 groups (each group has two spring pieces), the first and second natural frequencies of the high-frequency vibrating screen change little with K3, while the third natural frequency increases from 69.8 rad/s to 97.85 rad/s with K3. When the stiffness K3 reaches 1.31×e6 N/m, the third-order natural frequency reaches 78.33 rad/s, that is, changing the number of upper spring pieces has little effect on the low-order frequency of the vibrating screen system. Therefore, the amplitude and horizontal movement speed of the upper sieve can be controlled by changing the number of spring pieces by reasonable design, and the screening efficiency can be improved according to the tobacco feeding situation. when the number of spring piece groups is equal to 40 groups, resonance will occur at this time so that the mechanical equipment can work normally.  As shown in Fig. 5, when the stiffness is unchanged, the stiffness and are changed, that is, the number of elastic supports of ROSTA. It can be seen that the number of ROSTA elastic supports has little effect on the third-order natural frequency and the natural frequency increases as the number of elastic supports increases. When the number of elastic supports reaches 10, the screen body will resonate. The change of the number of lower spring pieces will affect the low-order natural frequency of the system. Considering the influence of the quality of the lower screen body, a reasonable design can effectively reduce the large lateral vibration of the vibrating screen when the machine is switched on and off.
As shown in Fig. 6, when the stiffness does not change, changing the magnitude of the vibration mass m1, that is, changing the mass of the intermediate frame, the range of the third-order natural frequency changes little, but it decreases as the mass of the intermediate frame increases. When = 816 kg, the screen body generates resonance. From calculation and analysis, the mass of the lower sieve body is reasonable between 600 and 1000 kg. The change of the lower sieve body mass has little effect on the vibration characteristics of the system, but if its mass is too small, it will cause a larger side directional vibration of the vibrating machine which will affect system stability.

Results and discussion
Using data acquisition and signal processing equipment (DASP), with 12 acceleration sensors and force hammers, the motion characteristics of the upper and lower screen bodies of the dual-mass vibrating screen were measured, as shown in Fig. 8. The measurement is divided into two parts. First, the amplitudes of different points on the upper and lower sieve bodies are measured to study the basic motion characteristics. Second, the motion characteristics of the upper and lower sieves are analyzed and studied through the measured images and specific amplitude data. and the effect of different numbers of spring pieces on the motion characteristics is further studied. The relationship between the horizontal and vertical vibration amplitudes of the upper and lower screen bodies and the vibration characteristic parameters of the screen shaker are studied. Using force hammer sensor and 12 acceleration sensors, using the MIMO method, through the DASP modal analysis software, the rigid body modal analysis of the double-layer vibrating screen is carried out, and the elastic body modal analysis of the upper and lower screen bodies is carried out respectively.
Using a force hammer sensor and 12 acceleration sensors in Fig. 8, using the MIMO method, the DASP modal analysis software was used to perform rigid body modal analysis on the double-layer vibrating screen, and the elastic modal analysis was performed on the upper and lower sieve bodies.

Measurement of motion characteristics
The DASP instrument was used to measure a type of high-frequency double-mass vibrating screen in the laboratory. Select two points on the lower sieve body, named point 1 and point 2; select two points on the upper sieve body, named point 5 and point 6. The arrangement of measuring points is shown in Fig. 9. The time-domain waveform analysis diagram of these four points measured by the DASP instrument is shown in Fig. 9. Where Fig. 9(a) shows the motion waveform of four points in the horizontal direction and Fig. 9 According to the actual situation, the mass of the upper sieve of the dual-mass high-frequency vibrating screen we measured is 200 Kg; the mass of the lower sieve is 976 Kg (including the motor), and the motor that provides the exciting force for the entire vibrating screen is located in the lower sieve, which makes the mass of the lower sieve larger than the mass of the upper sieve. According to the momentum conservation and the derived Eq. (4) obtained in the first part, it is found that the momentum conservation relationship is satisfied when the measured waveform amplitude and the vibration quality of the upper and lower tanks are brought into the momentum conservation Eq. (4) for calculation, which verifies the correctness of the theoretical research in the first part. The waveform diagram of Fig. 10 is sorted out and summarized to obtain a correspondence table between test points and amplitudes, as shown in Table 2.
According to the table of horizontal and vertical amplitudes of the working state, it can be concluded that the amplitudes of the points on the upper and lower sieve bodies have different positive and negative amplitudes at the same time, which means that the motion is reversed, which probably meets the law of conservation of momentum.
The four corners of the lower sieve body are selected as test points, and the displacement waveform in the horizontal direction is measured, as shown in Fig. 11.  Fig. 11, we can see that the motion of the sieve body is periodic, and the vibration amplitudes of the four corner points are relatively similar, and the vibration periods are consistent, indicating that the relative vibration of the upper and lower grooves can be simplified into their respective rigid body vibration.

Effect of spring plate number on motion characteristics
The different number of spring leaves influences the motion characteristics of the screen body of the vibrating screen. Therefore, it is an important research direction of this paper to make the motion of the screen body controllable by changing the number of spring leaves.
To ensure the safe and stable operation of the vibrating screen, we change the number of spring leaf groups used by the vibrating screen and use the DASP to measure the motion waveform of the vibrating screen under different spring leaf groups to obtain the required research data.
To ensure the normal and safe operation of the vibrating screen, we have set 20 sets of spring pieces, 19 sets of spring pieces, and 18 sets of spring pieces, and measured the movement of the points on the upper and sieve bodies in three cases, and obtained the corresponding relationship. The four test points above the lower sieve body are named point 1, point 2, point 3, and point 4; the four points above the upper sieve body are point 5, point 6, point 7, and point 8. The movement of these 8 points in horizontal and vertical directions was measured by the DASP instrument. Take the motion waveform diagram of each point under the normal operation of 20 groups of springs as an example, as shown in Fig. 12. Fig. 12(a) shows the movement of four measuring points on the lower sieve body under normal operation of 20 sets of spring pieces, and Fig. 12(b) shows the movement of four measuring points on the upper sieve body. After the motion stabilized, the amplitude of the 8 points in the horizontal direction was tested. In the same way, the horizontal motion of 8 points in the two cases of 19 sets of spring pieces and 18 sets of spring pieces is obtained, and the 1, 2, 3, and 4 points are located in the lower trough; the 5, 6, 7, and 8 points are located upper trough. Table 3   From Table 3, it can be concluded that for the points on the lower sieve body, the number of spring sheets has little effect on the points above. Considering the unavoidable errors in the actual operation, the amplitude of 19 groups of spring sheets is larger than that of 18 groups of spring sheets; the amplitude of three groups of spring sheets will fluctuate due to the specific position of the test point. As the number of upper spring pieces decreases, its rigidity also decreases. The amplitude of the upper screen body of the 18 groups of spring pieces is higher than that of the other two groups, and the third-order natural frequency of the system decreases, which is closer to the operating frequency of the motor 12.5 Hz, so the amplitude is larger. Fig. 13 shows the frequency response curves of the sensors located at the upper and lower troughs. The sensors are arranged horizontally and the hammer is excited horizontally. The horizontal vibration frequencies of the upper and lower vibrating screens are shown in Table 4 below. The measured natural frequency values are consistent. The 20, 19, and 18 sets of spring sheets are tested in the same way. The natural frequencies are shown in the Table 4.  The vibration amplitude along the vertical direction of the spring plate corresponding to the position of the connecting upper and lower grooves is studied. As shown in Fig. 14, the position of the upper groove is 38.39 m/s 2 , and the acceleration of the position of the lower groove is 6.96 m/s 2 , which is a 5.5:1 relationship. The mass ratio of the upper and lower grooves is about 1:5, so the vertical direction along the spring plate also basically meets the law of momentum conservation.

Rigid body modal test and parameter analysis of double-layer vibrating screen machine
For the lower sieve body, it can be seen from the measured data that the change of the number of spring plate groups has a greater impact on the motion amplitude of the upper sieve body, and with the increase of the number of spring plate groups, the amplitude of the measuring points on the upper sieve body as a whole has a decreasing trend. With the increase in the number of spring plates and the increase of the overall stiffness, the vibration amplitude will decrease. Increasing or decreasing the number of spring plate has a slight effect on the vibration frequency of the upper and lower tanks.

Vibrating screen modal test
12 piezoelectric acceleration sensors and a force hammer were used to attain the rigid body modal of the double-layer vibrating screen by the MIMO hammering method, as shown in Fig. 15.
The frequency of the motor in normal operation is 12.5 Hz. The modal testing frequency of the elastomer in the lower groove of the motor is shown in Table 5.
Because the upper groove body will produce a strong wind whistle when the machine is running, it is necessary to carry out an in-plane modal analysis. The results show that the local relative stiffness of the design is small, warping, and twisting vibration modes will occur. The modal analysis is shown in Fig. 16, The overall stiffness of the upper groove body needs to be increased or strengthened locally. The practice has proven that the follow-up has played an optimization and improvement effect on the design of the vibrating screen in Hangzhou Tobacco Factory. In addition to the relative reverse motion during operation, the upper and lower grooves are shown in Fig. 17(a), and there is relative torsional vibration outside the plane, as shown in Fig. 17(b), but the amplitude is small, and its influence should be considered in the structural design and selection of the spring sheets.  The picture below shows a vibrating screen machine placed on the second floor of an industrial plant in Hangzhou. It is required to minimize the vibration of the floor when the machine is working. Therefore, we control the mass ratio of and , reasonable choosing elastic support K1 and adjusting the number of spring pieces to control K2 according to the law of influence of the above parameters on the vibrating screen system to make the machine can operate normally at 12.33 Hz while greatly reducing the vibration to the floor.  When the double-layer vibrating screen is working normally, by comparing the maximum displacement and velocity amplitude between the upper and lower screen bodies of the vibrating screen and the floor, it can be found that the design requirements are well met.

Conclusions
Based on the above research on the motion characteristics of the sieve body and the study of the motion characteristics of the vibrating screen, we get some conclusions about the controllability of the vibrating screen.
The power source of the double-layer high-frequency vibrating screen machine is the key part of the controllable vibration, and the power comes from the motor installed on the lower screen body. The eccentric rotor of the two motors rotates reversely so that the vibrating screen obtains an exciting force to vibrate. The mass of the eccentric rotor in the motor, the angular velocity of rotation, and the radius of gyration of the rotor are three key quantities that affect the vibration characteristics of the vibrating screen. By increasing the mass of the eccentric rotor in the motor, the angular velocity of rotation and the radius of gyration, the vibration amplitude of the screen body of the vibrating screen can be increased.
The parameters of the double-layer high-frequency vibrating screen itself are also the influencing factors of controllability. The stiffness of the spring piece selected for the vibrating screen directly affects the vibration amplitude of the screen body. From the test data shown in Table 3, it is can be known that the greater the stiffness of the spring piece used, the smaller the vibration amplitude of the screen body. However, the increase or decrease of the number of spring piece has a limited influence on the vibration amplitude, which will affect the vibration frequency domain parameters of the upper and lower troughs. The selection of the number of spring pieces should also consider the overall vibrating quality.
The quality of the material placed in the vibrating screen is also a factor that affects the screen of the vibrating screen. According to the actual screening requirements, the appropriate quality of materials should be placed so that the screening requirements can be met. The relative vibration amplitude of the upper and lower tanks is mainly determined by the relative mass ratio and the number and stiffness of the spring pieces.
Through the experimental measurement, the overall motion of the vibrating screen machine in the project roughly meets the principle of conservation of momentum. We verified the correctness of the derivation results in the first part.
Through research, we have obtained the controllable design of the upper and lower sieve bodies of the double-layer high-frequency vibrating screen machine, which should be considered from the three aspects: the motor, the vibrating screen machine, and the material. The specific design should refer to the following conclusions: 1) The rotor in the motor affects the vibration of the screen. Increasing any one of the masses of the rotor, angular velocity and the radius of gyration will expand the vibration amplitude of the screen. The relative vibration amplitude of the upper and lower troughs mainly depends on their relative mass ratio and the number and stiffness of the spring pieces.
2) The number and performance of the rigid plate used by the vibrating screen itself affect the vibration characteristics of the screen body. The more spring sheets are used, the greater the stiffness of the spring sheets, and the smaller the vibration amplitude of the screen body. The increase or decrease of the number of spring sheets has a limited influence on the vibration amplitude, which will affect the vibration frequency domain parameters of the upper and lower troughs. The selection of the number of spring sheets should still consider the overall vibrating quality.
3) The material and the falling height of the material placed in the vibrating screen will affect the vibration of the screen body, and it should be controlled according to the specific screening requirements. The subsequent effects can be accurately tested and simulated.
4) The experimental results show that the overall motion of the vibrating screen in engineering, including horizontal, vertical, and vertical directions along with the spring pieces, approximately meets the law of momentum conservation, and can be used as an important reference factor for design.