Dynamics of a vibratory screening conveyor equipped with a controllable centrifugal exciter

2.1. 3D-model of the vibratory screening conveyor and the enhanced centrifugal exciter

The considered vibratory screening conveyor (Fig. 1) consists of the working member 1, vibration isolators (cylindrical coil springs) 2, and electromechanical exciter 3. The working member 1 contains the lower conveying tray (chute) and the upper rod-type screen.

Fig. 1General design of the vibratory screening conveyor and the enhanced centrifugal exciter

The enhanced vibration exciter (Fig. 1) consists of the base 4, on which all the elements are installed. The servo motor 5 sets the belt transmission 6 into motion. The driven pulley is fixed on the main shaft 14 installed in the bearing support 13. The use of the linear (guide, pilot) ball bearing 15 allows for changing the angular position of the rocker 16 and the eccentricity of the unbalanced mass 18. The passive balancing mechanism 17 is applied to ensure the zero-value inertial parameters of the exciter in its folded position, providing zero centrifugal force. The active eccentricity control system consists of the step motor 7, jaw clutch 8, and lead (screw) shaft 9 installed in the bearing support 11. The spring-loaded nut 10 is fixed on the sliding plate installed on the guide (pilot) bearing 15. In order to define the end position of the sliding plate, the inductive sensor 12 is used. The operation of the considered centrifugal exciter can be described as follows. The servo motor’s shaft sets the unbalanced mass into the rotary motion through the belt transmission. The unbalanced mass eccentricity controlled by the step motor defines the magnitude of the centrifugal force, which periodically changes its direction and disturbs (excites) the oscillations of the conveyor’s working member.

2.2. Dynamic diagram and mathematical model of the conveyor’s oscillatory system

The simplified dynamic diagram of the conveyor’s oscillatory system is shown in Fig. 2. The working member is modeled as a rigid body of the mass $m_1$. The body is suspended from the stationary base by the system of spring-damper elements, characterized by the stiffness coefficients $k_{1x}=0.5k_x$, $k_{1y}=0.5k_y$ and viscous friction (damping) coefficients $c_{1x}=0.5c_x$, $c_{1y}=0.5c_y$. The oscillations are excited by the unbalanced rotor hinged to the body at the point $E$. The angular speed (forced cyclic frequency) is assumed constant and equal to $\omega$. In the simplified diagram, the control system regulating the position (eccentricity $\epsilon$) of the unbalanced mass $m_2$ is omitted. Let us consider the case when the eccentricity $\epsilon$ and forced cyclic frequency $\omega$ are the controllable parameters allowing for adjusting the system operational parameters (amplitudes, velocities, accelerations) in accordance with the technological requirements.

A mathematical model describing the working member oscillations can be deduced using the Euler-Lagrange equations:

where the dots above the corresponding quantities denote the first-order or the second-order derivative of this quantity with respect to time.

Fig. 2Dynamic diagram of the conveyor’s oscillatory system

3.1. Numerical modeling of the system oscillations in the Mathematica software

Let us study the conveyor’s oscillations by numerical solving of the system of differential equations (1) in the Mathematica software with the help of the Runge-Kutta methods. The following input parameters are considered: $m_1=$ 56.072 kg, $m_2=$ 0.301 kg, $c_x=$ 180 (N∙s)/m, $k_x=$2.149∙10⁴ N/m, $\omega =$314 s^-1, $\alpha =$ 30°. The numerical modeling is carried out for the following values of the unbalanced mass eccentricity $\epsilon$: 0, 10, 20, 35 mm. The results are shown in Fig. 3 by the blue, brown, green, and red lines, respectively. Horizontal and vertical amplitudes vary from zero at $\epsilon =$0 mm to 0.1 mm and 0.16 mm, respectively, at $\epsilon =$35 mm. The maximal values of the corresponding speeds reach 0.03 m/s and 0.05 m/s. Horizontal acceleration is about 9 m/s², while the vertical one exceeds the value of 16 m/s². This allows for concluding about the detachable motion conditions when the parts being conveyed are jumping (bouncing) over the conveying (screening) surface. In addition, the obtained results substantiate the initial idea of this research: by controlling the eccentricity of the unbalanced mass, it is possible to adjust the machine’s operational conditions in accordance with the technological requirements set for different materials to be conveyed and sieved. Considering the case of $\epsilon =$10 mm, the amplitude values of the horizontal and vertical displacements are about 0.03 mm and 0.05 mm, speeds – 0.009 m/s and 0.015 m/s, accelerations – 2.5 m/s² and 4.5 m/s². At $\epsilon =$20 mm, the corresponding values of the system’s kinematic parameters are the following: 0.06 mm and 0.09 mm, 0.018 m/s and 0.03 m/s, 5 m/s² and 9 m/s².

Fig. 3Numerically modeled kinematic characteristics of the conveyor’s oscillatory system

3.2. Simulation of the conveyor operation in the SolidWorks software

Let us verify the correctness of the obtained numerical modeling results by computer simulation (virtual experiment) of the conveyor operation with the help of the SolidWorks Motion software. In the considered case, let us adopt the eccentricity of $\epsilon =$35 mm. The corresponding simulation model and results are presented in Fig. 4. All the parameters of the considered oscillatory system are similar to those mentioned above. Unlike the numerical modeling, the simulation in the Solidworks Motion software allows for taking into account the simplified operational parameters of the electric drive used for actuating the oscillatory system. The transient (starting) conditions last for about 0.8 s. Analyzing the obtained simulation results, the conclusions about their satisfactory agreement with the numerical modeling results can be drawn. The amplitude values of the horizontal and vertical displacements are about 0.1 mm and 0.16 mm, speeds – 30 mm/s and 51 mm/s, accelerations – 9100 mm/s² and 16000 mm/s².

The experimental prototype of the considered vibratory screening conveyor (Fig. 1) has been manufactured at the Vibroengineering Laboratory of Lviv Polytechnic National University and is currently being tested under different operational conditions. Further research on this subject is focused on developing the automatic control system adjusting the conveyors’ excitation parameters (forced frequency and unbalanced mass eccentricity) in accordance with the operational conditions (effective (useful) load, conveying speed, overloading factor, etc.) and technological requirements set for different types of materials to be conveyed and sieved.

Fig. 4Simulation model and results of virtual experiments performed in the SolidWorks software